Steady-state Analytical Solutions Research Articles

SALINITY BUILD-UP DUE TO BRINE DISCHARGES INTO SHALLOW COASTAL WATERS

A two-dimensional mathematical model simulating the long-term build-up of salinity around the brine outfall is developed in this paper. A flat seabed and steady longshore current are assumed as a first step towards modelling the actual scenario. Transient and steady-state analytical solutions for continuous point source condition are derived. These solutions are then applied to assess shoreline/offshore discharge. Characteristic features such as the temporal buildup of salinity excess and the variation of the shoreline salinity with respect to time and outfall location are investigated. The effect of the advection-to-diffusion ratio ν in the modelling results is also discussed.

Existence of Bounded Discrete Steady-State Solutions of the Van Roosbroeck System on Boundary Conforming Delaunay Grids

The classic van Roosbroeck system describes the carrier transport in semiconductors in a drift diffusion approximation. Its analytic steady state solutions fulfill bounds for some mobility and recombination/generation models. The main goal of this paper is to establish the identical bounds for discrete in space, steady state solutions on 3d boundary conforming Delaunay grids and the classical Scharfetter-Gummel scheme. Together with a uniqueness proof for small applied voltages and the known dissipativity (continuous as well as space and time discrete), these discretization techniques carry over the essential analytic properties to the discrete case. The proofs are of interest for deriving averaging schemes for space or state dependent material parameters, which preserve these qualitative properties, too. To illustrate the properties of the scheme, 1, 4, 16 elementary cells of a modified CoolMOS-like structure are depleted by increasing the applied voltage until steady state avalanche breakdown occurs.

A simplest steady-state Munch-like model of phloem translocation, with source and pathway and sink

In the 80 years since its introduction by Münch, the pressure-driven mass-flow model of phloem translocation has become hegemonic, and has been mathematically modelled in many different fashions but not, to our knowledge, by one that incorporated the equations of hydrodynamics with those of osmosis and slice-source and slice-sink boundary conditions to yield a system that admits of an analytical steady-state solution for the sap velocity in a single sieve tube. To overcome this situation, we drastically simplified the problem by: (i) justifying a low Peclet number idealisation in which transverse variations could be neglected; (ii) justifying a low viscosity idealisation in which axial pressure drops could be neglected; and (iii) assuming a sink of strength sufficient to lower the photosynthate concentration at the extreme distal end of the sieve tube to levels at which it became unimportant. The resulting ordinary nonlinear second-order differential equation in sap velocity and axial position was of a generalised Liénard form with a single forcing parameter; and this is reason enough for the lack of a known analytic solution. However, since the forcing parameter was very large, it was possible to deduce approximate second-order solutions for behavior in the source, sink and transport regions: the sap velocity is zero at the slice-source, climbs with exponential rapidity to a plateau, maintains this plateau over most of the sieve tube, and then drops with exponential rapidity to zero at the slice-sink.

Analytical decoupling of poroelasticity equations for acoustic-wave propagation and attenuation in a porous medium containing two immiscible fluids

Poroelasticity theory has become an effective and accurate approach to analyzing the intricate mechanical behavior of a porous medium containing two immiscible fluids, a system encountered in many subsurface engineering applications. However, the resulting partial differential equations in the theory intrinsically take on a coupled form in the terms pertinent to inertial drag, viscous damping, and applied stress, making it difficult to obtain closed-form, steady-state analytical solutions to boundary-value problems except in special cases. In the present paper, we demonstrate that, for dilatational wave excitations, these partial differential equations can be decoupled analytically into three Helmholtz equations featuring complex-valued, frequency-dependent normal coordinates that correspond physically to three independent modes of dilatational wave motion. The normal coordinates in turn can be expressed in the frequency domain as three different linear combinations of the solid dilatation and the linearized increment of fluid content for each pore fluid, or equivalently, as three different linear combinations of total dilatational stress and two pore fluid pressures. These representations are applicable to strain-controlled and stress-prescribed boundary conditions, respectively. Numerical calculations confirm that the phase speed and attenuation coefficient of the three dilatational waves represented by the Helmholtz equations are exactly identical to those obtained previously by numerical solution of the dispersion relations for dilatational wave excitation of a porous medium containing two immiscible fluids. Thus, dilatational wave motions in unsaturated porous media subject to suitable boundary conditions can now be accurately modeled analytically.

Mass-losing accretion discs around supermassive black holes

We study the effects of outflow/wind on the gravitational stability of accretion discs around supermassive black holes using a set of analytical steady-state solutions. Mass-loss rate by the outflow from the disc is assumed to be a power-law of the radial distance and the amount of the energy and the angular momentum which are carried away by the wind are parameterized phenomenologically. We show that the mass of the first clumps at the self-gravitating radius linearly decreases with the total mass-loss rate of the outflow. Except for the case of small viscosity and high accretion rate, generally, the self-gravitating radius increases as the amount of mass-loss by the outflow increases. Our solutions show that as more angular momentum is lost by the outflow, then reduction to the mass of the first clumps is more significant.

Analytical solution of the steady-state molecular-motor-assisted transport equations governing distribution of intracellular particles within a cell dendrite

A steady-state analytical solution is obtained for a one-dimensional problem describing transport of intracellular organelles in a cell arm (a dendrite or axon). The model of intracellular trafficking of organelles (intracellular particles) is based on molecular-motor-assisted transport equations suggested by Smith and Simmons [D.A. Smith, R.M. Simmons, Models of motor-assisted transport of intracellular particles, Biophysical Journal 80 (2001) 45–68.]. According to this model, a fraction of intracellular space is occupied by filaments (microtubules); the remaining space allows for diffusion of unbound particles. Depending on the direction of particle transport provided by molecular motors running along these microtubules, the microtubules are divided into inward and outward microtubules. Unbound (free) organelles may bind to microtubules; organelles attached to microtubules may detach from them; the binding/detachment kinetic processes are specified by the first rate constants.

A Fourier Series-Based Analytical Solution for Three-Dimensional Conjugate Heat Transfer Problems in Microprocessor Cooling

A fully analytical steady state solution is presented for the problem of 3-D conjugate heat transfer in flip chip electronic packages with multiple general areal power sources, cooled by heatsinks using forced convection. The inherent simplicity of the proposed approach allows for expedient yet detailed package cooling/thermal analysis and evaluation of what-if scenarios while requiring only modest computational resources such as a personal computer. The soundness of the approach presented is demonstrated via excellent correlation with the results of experimental tests on flip-chip thermal test packages in a wind tunnel. The techniques developed herein are fairly general and can be readily adapted to other package designs wherein the primary heat transfer path is through the die, die-lid interface, lid, lid-sink interface and heatsink to the airflow through the fins.

Analytical Steady-State Solution of Single-Probe Tests in a Horizontal Well and Its Application to Estimate Horizontal and Vertical Permeabilities

Summary During a single-probe test in a vertical well, the probe of a formation tester is set horizontally against a sidewall of the borehole. In a horizontal well, the probe can be set either horizontally against a sidewall or vertically against the top or bottom wall of the borehole. When the probe is set horizontally against a sidewall, the solution for a horizontal well is the same as that for a vertical well, if the wellbore curvature is not considered (in other words, the wellbore radius is considered to be much larger than the probe radius). However, when the probe is set vertically against the top or bottom wall, the solution for a horizontal well is not available in the literature. This paper presents an analytical solution to the described problem. The solution shows that the relationship between the pressure drop and flow rate for a single-probe test in a horizontal well depends on the geometric average of horizontal and vertical permeabilities when the probe is set vertically against the borehole. One of the important applications of the solution is to estimate horizontal and vertical permeabilities from two probe tests. If two tests are conducted at the same measured depth, with the probe set vertically in one test and horizontally in the other, two permeabilities are obtained. Then the horizontal and vertical permeabilities at the measured depth can be estimated by combining these two permeabilities. The analytical solution is validated by numerical simulation, and the solution is shown to be the same as a published analytical solution for a vertical well in an isotropic formation. Examples to estimate horizontal and vertical permeabilities from synthetic tests are presented.

COST ANALYSIS OF THE R-UNRELIABLE-UNLOADER QUEUEING SYSTEM

This paper analyzes the unloader queueing model in which N identical trailers are unloaded by R unreliable unloaders. Steady-state analytic solutions are obtained with the assumptions that trip times, unloading times, finishing times, breakdown times, and repair times have exponential distributions. A cost model is developed to determine the optimal values of the number of unloaders and the finishing rate simultaneously, in order to minimize the expected cost per unit time. Numerical results are provided in which several steady-state characteristics of the system are calculated based on assumed numerical values given to the system parameters and the cost elements. Sensitivity analysis is also studied.

Nanodomain Coupling between Ca2+ Channels and Ca2+ Sensors Promotes Fast and Efficient Transmitter Release at a Cortical GABAergic Synapse

It is generally thought that transmitter release at mammalian central synapses is triggered by Ca2+ microdomains, implying loose coupling between presynaptic Ca2+ channels and Ca2+ sensors of exocytosis. Here we show that Ca2+ channel subunit immunoreactivity is highly concentrated in the active zone of GABAergic presynaptic terminals of putative parvalbumin-containing basket cells in the hippocampus. Paired recording combined with presynaptic patch pipette perfusion revealed that GABA release at basket cell-granule cell synapses is sensitive to millimolar concentrations of the fast Ca2+ chelator BAPTA but insensitive to the slow Ca2+ chelator EGTA. These results show that Ca2+ source and Ca2+ sensor are tightly coupled at this synapse, with distances in the range of 10-20 nm. Models of Ca2+ inflow-exocytosis coupling further reveal that the tightness of coupling increases efficacy, speed, and temporal precision of transmitter release. Thus, tight coupling contributes to fast feedforward and feedback inhibition in the hippocampal network.

Modeling and stability analysis for an ultrasonic motor

The development of motor models, adequate for motor design and optimization is still behind the technological state of the art. The reason for this lies in the fact that a couple of years ago the main emphasis regarding ultrasonic motors was on finding new principles or prototypes while today this has shifted towards optimization of known working concepts and motors. A systematic motor optimization demands, among at her things, a detailed dynamical understanding of the motor which not too often exists for ultrasonic motors in general.The present study deals with modeling a wobbling disk piezoelectric motor. For the model the analytic steady state solution is derived and stability of the solution is investigated. The model and the analytic steady state solution yield a detailed insight in essential dynamical phenomena by which the motor is driven. The model framework may be a guideline providing understanding in the working principle, dynamics, weighting of the parameters with respect to reliable, stable operation and optimization of such motors.

Quantification of infiltration processes in urban areas by accounting for spatial parameter variability

As a consequence of human living and activity, water infiltration to the urban subsurface occurs from a variety of different sources, like precipitation, irrigation, leaking pipes and sewers, septic tanks and rainwater infiltration ponds. This infiltration is strongly related with quality issues of the infiltrated water and further impact on groundwater quality. In order to set up an integrated urban water balance it becomes essential to estimate the infiltration processes, i.e. water flow and solute transport, from these different infiltration sources and to take into account the large spatial variability of sediment properties, the geometric settings of these sources and the groundwater table. For that purpose, the development of simple, physically-based quantification approaches is required in order to establish an efficiently working prediction and risk analysis tool within the framework of an integrated urban water management system. The scope of the presented work was to demonstrate the applicability of the developed approaches at urban scale. Since a detailed, three-dimensional, numerical quantification of the infiltration processes within the entire urban area is not possible, the individual sources were considered as independent within the EU AISUWRS project. Different models were developed for balancing infiltration from areal and point sources with respect to the related flow pattern. The analytical model UL_FLOW, based on one-dimensional, steady state analytical solutions, allows the estimation of conservative tracer residence times in layered sediments under varying infiltration rates. The numerical model WSTM, based on a three-dimensional random walk approach, calculates water and solute transport from pipe leaks. Additionally, the sources were classified in accordance to the spatial distribution of the parameters determining the infiltration processes. UL_FLOW was applied to data sets from the city of Rastatt within a case study of the AISUWRS project. Each neighbourhood of water balance computation by the Urban Volume and Quality Model (UVQ) was defined as an areal infiltration source with unique parameter values for sediment depth, profile and properties, as well as infiltration rate time series. Groundwater recharge and residence time series were computed for each neighbourhood. Relevant statistical parameters obtained by time series analyses from those time series could be mapped by GIS. Point infiltration, particularly from sewers, was classified due to the sediment parameters and the distance to the groundwater table at each source location in order to reduce computational efforts. WSTM computations provided time series of groundwater recharge and tracer breakthrough for some specific cases. The analytical model UL_FLOW provides fast and efficient computation of groundwater recharge and residence times accounting for storage effects within the unsaturated zone of urban areas. The reliability of this model has been shown by cross validation with HYDRUS1D. Because of the high computational effort, WSTM could provide only short-term simulations for some specific parameter sets for which residence time estimates could be derived. UL_FLOW provides an analytical modelling tool for balancing one-dimensional areal infiltration and estimating residence times under varying conditions including spatial parameter variability. These balances could be used for assessing the impact of those infiltration sources on groundwater quality. The tracer breakthrough from point infiltration sources computed by WSTM could also be used for such kinds of assessment. The larger spatial parameter variability associated with these sources could be handled by classification in GIS environments. Similar to the areal sources, a simple balance approach for point sources based on analytical solutions needs to be developed for estimating residence times in order to avoid large computational efforts. Such a model would complete the balancing of all kinds of infiltration sources in urban areas efficiently. Since the approaches are based on the balance of the physical processes, they have a large predictive capability and could be included into an integrated urban water balance and management system. The mapping of the statistical values of the residence times provides a tool to compare parts of the urban areas and to visualize differences between urban water management scenarios.

Asymptotic models of solidification in cooling thin-layer flows of a highly viscous fluid

Asymptotic models are constructed for the solidification process in a highly viscous film flow on the surface of a cone with a given mass supply at the cone apex. In the thin-layer approximation, the problem is reduced to two parabolic equations for the temperatures of the liquid and the solid coupled with an ordinary differential equation for the solidification front. For large Peclet numbers, an analytical steady-state solution for the solidification front is found. A nondimensional parameter which makes it possible to distinguish flows (i) without a solid crust, (ii) with a steady-state solid crust, and (iii) with complete solidification is determined. For finite Peclet numbers and large Stefan numbers, an analytical transient solution is found and the time of complete flow solidification is determined. In the general case, when all the governing parameters are of the order of unity, the original system of equations is studied numerically. The solutions obtained are qualitatively compared with the data of field observations for lava flows produced by extrusive volcanic eruptions.

Impact of Coastal Land Reclamation on Ground Water Level and the Sea Water Interface

Land reclamation in coastal areas may have a significant effect on local ground water systems. Steady-state analytic solutions based on Dupuit and Ghyben-Herzberg assumptions are derived to evaluate this effect. Two situations are considered, both with ground water flow resulting from precipitation recharge: the coastal aquifer of an extensive landmass and an island. The results show that after reclamation, the water table rises and the salt water-fresh water interface moves seaward. The degree of these changes depends on the extent of reclamation and the hydraulic conductivity of the fill material. For the island situation, the reclamation displaces the ground water divide and changes the ground water conditions in the entire island. An unintended advantage of the reclamation is an increase of fresh ground water resource because the reclaimed land can be an additional aquifer and rain recharge takes place over a larger area.

Advective and diffusive contributions to reactive gas transport during pyrite oxidation in the unsaturated zone

Pyrite oxidation in unsaturated mine waste rock dumps and soils is limited by the supply of oxygen from the atmosphere. In models, oxygen transport through the subsurface is often assumed to be driven by diffusion. However, oxygen comprises 23.2% by mass of dry air, and when oxygen is consumed at depth in the unsaturated zone, a pressure gradient is created between the reactive zone and the ground surface, causing a substantial advective air flow into the subsurface. To determine the balance between advective and diffusive transport, a one‐dimensional multicomponent unsaturated zone gas transport model is developed. Both advection‐diffusion and Maxwell‐Stefan model formulations are presented. A steady state analytical solution is derived that provides insight into solution behavior and which can be used to test numerical models. A numerical solution is obtained for both the steady and transient cases. At steady state, advection comprises approximately 23% of the total oxygen flux, with the contribution of advection being almost entirely determined by the composition of the atmosphere. Other parameters, such as the permeability, have a negligible effect on the proportion of advective flows at steady state. However, greater pressure gradients are found in low‐permeability soils. In transient cases, advective fluxes depend on the initial conditions and can be far greater than diffusive fluxes. In contrast to steady state conditions the transient case is sensitive to other model parameters; for example, the time to approach steady state depends exponentially on the distance between the soil surface and the subsurface reactive zone.

A parallel adaptive barotropic model of the atmosphere

The parallel adaptive model PLASMA has been developed for modeling a barotropic atmosphere. This model adapts the computational grid at every time step according to a physical error indicator. Thus, compared to uniform grid experiments the number of grid points is reduced significantly. At the same time, the error increases only slightly, when comparing with uniform grid solutions. For the discretization of the underlying spherical shallow water equations a Lagrange–Galerkin method is used. The unstructured triangular grid is maintained by the grid generator amatos and the large linear systems are solved by the parallel solver interface FoSSI. Experimental convergence is shown by means of steady-state and unsteady analytical solutions. PLASMA yields satisfactory results for quasi standard experiments, that is the Rossby–Haurwitz wave and zonal flows over an isolated mountain.

Numerical and Analytical Modelling of the Gas Well Liquid Loading Process

Summary Liquid loading is a serious problem in areas where gas fields are maturing. This paper presents an analysis of the production behavior of liquid-loaded wells over time. This clearly shows that these wells can operate at two different rates—a stable rate, at which full production is taking place, and at a lower metastable rate, at which liquid-loading effects play a role. A model has been constructed that enhances the understanding of the process of water buildup and drainage in gas wells. It assumes a single gas- and water-coproduction point and a single water reinjection point. As expected, a water column is built up in the well as soon as production takes place below the critical rate. As observed in the field, for good inflow performance, a metastable flow rate can be observed. At this state, the water-reinjection and water-coproduction rate are equal to one another, and the water-column height stabilizes. A sensitivity analysis has been carried out to determine how well parameters influence the metastable flow rate, the time required to reach this metastable rate, the corresponding water-column height, and the shut-in time required to drain this water column. The results of the analysis indicate that significant metastable flow rates occur in wells that have good inflow performance, a low water/gas ratio, and a large distance between injection and production point. Furthermore, a steady-state analytical solution has been derived for the metastable rate and stabilized water-column height confirming the numerical-analysis results. Introduction Many of the mature offshore gas fields in the southern North Sea have already experienced considerable pressure depletion resulting in significantly reduced gas flow rates and, eventually, liquid loading. Insight into the flow characteristics of these liquid-loaded wells will help manage tail-end production (i.e., will help define the most effective means of accelerating and maximizing their ultimate recovery.

Light transport model in a n-layered mismatched tissue

The steady-state and time-domain analytical solutions of the diffusion equation for photon migration through a highly scattering n-layered mismatched medium have been obtained. The effect of the refractive-index mismatch is taken into account, and the extrapolated boundary condition has been considered. A Monte Carlo code for photon migration through an n-layered mismatched medium has also been developed. Comparisons with the results of Monte Carlo simulations showed that the analytical solutions correctly describe the photon migration, while reflectance is observed.

Radon exhalation from phosphogypsum building boards: symmetry constraints, impermeable boundary conditions and numerical simulation of a test case

Comprehensive understanding of 222Rn exhalation from phosphogypsum-bearing building material and its accumulation in indoor air is likely to rely on numerical simulation, particularly if transient effects, three-dimensional domains and convection are to be included and investigated. Yet, experimental data and analytical results are helpful (if not crucial) as far as validation is concerned. Having in mind computational code simplicity and in the light of a recent experimental and theoretical report on 222Rn release from phosphogypsum boards for housing panels, this paper presents and discusses an alternative testing set-up and the corresponding boundary conditions, namely one side of the panel bounded by impermeable wall. Although this is a new facility to be tested, the resultant steady-state one-dimensional diffusion-dominant analytical solution is shown to match the counterpart deduced in the aforementioned previous report, despite it relaxes the constraint of symmetry about the phosphogypsum board centerline, which is inferred in that prior experimental scenario. In addition, numerical results are conducted for a diffusion-dominant two-dimensional time-varying test case concerning 222Rn accumulation in a closed chamber having an exhaling phosphogypsum board tightly placed at one wall.

Optical bistability of a composite system induced by a broadband squeezed vacuum

A composite system consisting of a degenerate optical parametric oscillator (DOPO) and N two-level atoms interacting with a broadband squeezed vacuum (SV) centred at frequency ωs and an input monochromatic pumping field with a frequency ωp is analysed. The corresponding explicit analytical steady-state solutions in the central mode ωp = ωs are derived and the result displays optical bistability (OB). In addition, the influence of the broadband SV on the bistable behaviour is analysed in detail.